Computing general-relativistic effects from Newtonian N-body simulations: Frame dragging in the post-Friedmann approach

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Abstract

We present the first calculation of an intrinsically relativistic quantity, the leading-order correction to Newtonian theory, in fully nonlinear cosmological large-scale structure studies. Traditionally, nonlinear structure formation in standard $\mathrm{\ensuremath{\Lambda}}\text{CDM}$ cosmology is studied using N-body simulations, based on Newtonian gravitational dynamics on an expanding background. When one derives the Newtonian regime in a way that is a consistent approximation to the Einstein equations, the first relativistic correction to the usual Newtonian scalar potential is a gravitomagnetic vector potential, giving rise to frame dragging. At leading order, this vector potential does not affect the matter dynamics, thus it can be computed from Newtonian N-body simulations. We explain how we compute the vector potential from simulations in $\mathrm{\ensuremath{\Lambda}}\text{CDM}$ and examine its magnitude relative to the scalar potential, finding that the power spectrum of the vector potential is of the order ${10}^{\ensuremath{-}5}$ times the scalar power spectrum over the range of nonlinear scales we consider. On these scales the vector potential is up to two orders of magnitudes larger than the value predicted by second-order perturbation theory extrapolated to the same scales. We also discuss some possible observable effects and future developments.

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